Fourier spectroscopy for analyzing the composition of a sample frequently employs a two-beam interferometer such as a Michelson interferometer wherein an optical beam is divided by partial reflection into two separate wavefronts, one directed along a fixed-length arm and the other directed along a variable-length arm which is varied to cause spectral scanning. The basic elements of a conventional Michelson interferometer 10 are shown in simplified schematic diagram form in FIG. 1.
Incident electromagnetic radiation E.sub.in from a light source 12 is divided into two separate beams by a beamsplitter 14. One beam E.sub.rf is reflected to a fixed retroreflector 24 shown in FIG. 1 as a cube corner mirror. This reflected beam E.sub.rf is reflected back to the beamsplitter 14 where only one-half of its intensity is transmitted as a portion of the output beam E.sub.out. The second beam E.sub.tr is transmitted to a movable retroreflector 22 which reflects this radiation back to the beamsplitter 14 where one-half of its intensity is reflected to contribute the second portion of the output radiation, or beam, E.sub.out. The output radiation beam E.sub.out is provided to a detector 26 and appropriate signal processing circuitry which is not shown in the figure for simplicity.
The two equal portions of the incident electromagnetic wave E.sub.rf and E.sub.tr are passed along separate paths and are then recombined. By changing the position of the moveable retroreflector 22 relative to the fixed retroreflector 24 by means of displacement means 21, the recombined output beam E.sub.out will undergo constructive or destructive interference. The following equation defines the relationship between the wavenumber (or number of wavelengths per centimeter of the wave) and its frequency f. EQU f=c.sigma. (1)
For an infrared (IR) interferometer wavenumbers .sigma. are on the order of 1000 per cm, for measurements of light (having a speed c) , where ##EQU1##
This wavelength leads to a very high frequency (on the order of 10.sup.13 Hz). There are no practical methods for detecting wave amplitudes of these frequencies directly. Detectors sensitive to radiation of this frequency and wavelength detect the power density transmitted by the wave in a square law fashion. The physical principle for computing the power carried by the output beam E.sub.out may be computed by multiplying the wave amplitude function given by equation 2 with its complex conjugate. The conjugate of a complex number is itself a complex number with an equal real part and an imaginary part equal to the negative of the imaginary part of the number as given by the following ##EQU2##
The expression for the signal recorded by a square law detector measuring the power of the output radiation of the interferometer is derived by simplifying Eq. 3 to the following form ##EQU3##
The constant term (with respect to x) represents a bias power that is one-half of the incident power. The cosine term (as a function of x and .sigma.) exemplifies the application of the interferometer to spectroscopy (and other applications), For simplicity sake we will drop the constant term 1/8A.sub.in.sup.2, since it is not of interest in this discussion.
The practical use of the interferometer for spectroscopic applications involves the generation of a waveform by scanning the one retroreflector which has the effect of varying the retardation (x) over a continuous finite range of values. For a monochromatic light source, a single constant wavenumber .sigma. value, the scan will generate a cosine wave of amplitude P-detector and of frequency .sigma. , frequency being one over the distance in this case rather than one over time as in the usual case. Equation 4 expresses this behavior concisely. FIG. 2 is a graphic representation of equation 4 where detected power P.sub.D is plotted along the vertical axis and scanning mirror position, or retardation, x is plotted along the horizontal axis.
In most cases, it is not possible to support the beamsplitter 14 in a freestanding mode as shown in FIG. 1. The beamsplitter is typically comprised of a substrate which is soft, malleable and fragile. For the infrared, near infrared, or visible spectral regions the beamsplitter will generally take the form of a very thin film of dielectric material such as of germanium. The thin dielectric film must be deposited on a substrate of substantially greater thickness than the film, where the substrate is comprised of a material which is transparent over the spectral bandwidth of interest. In the case of the mid-infrared regions, potassium bromide (KBr) is a good choice for the substrate. A typical Michelson interferometer 20 constructed in this fashion is shown in simplified schematic diagram form in FIG. 3, where elements common to the spectrometer shown in FIG. 1 have been assigned the same identifying number. The difference between the interferometer 20 of FIG. 3 and interferometer 10 of FIG. 1 is the inclusion of substrate 28 in the former to support the thin film beamsplitter 14.
This modification does not affect the reflected beam path, as the previously discussed equations for the reflected beam path in the interferometer 10 of FIG. 1 still apply. The beam traversing the transmitted path E.sub.tr is, however, affected by the addition of substrate 28 to interferometer 20 shown in FIG. 3. The beam is refracted according to Snell's Law which bends the path. However, the effect is cancelled upon refraction at the beam exit surface of substrate 28. Beam refraction upon reentering substrate 28 following reflection by the scanning retroreflector 22 brings the transmitted beam E.sub.tr to the proper point for recombination just as in the case of the interferometer 10 of FIG. 1. The recombined output beam E.sub.out also undergoes refraction upon exit toward detector 26, however, the optical interactions of the recombined beam are not relevant to the derivation of the interferometric principles discussed herein and will therefore be ignored. The interactions of interest are the interactions of the electromagnetic radiation with the substrate 28 along the transmitted interferometer arm E.sub.tr.
The effect of substrate 28 is to increase the optical distance traversed by the transmitted beam E.sub.tr. The increased distance is defined as z.sub.str, which is a positive number and is given by the following expression: EQU Z.sub.str =Z.sub.tr +Z.sub.s ( 5)
The added path length of the transmitted beam E.sub.tr arises from two effects of transmission of an electromagnetic wave through a medium. First, Snell's Law predicts that the beam path will be bent upon refraction which increases the path slightly. Assuming a given fixed angle of incidence, such as the 45.degree. for the case of interferometer 20 shown in FIG. 3, the angle of refraction can be computed. Where the medium is air having an index of refraction n of 1, the calculation is simplified and is given by the following expression ##EQU4##
With the angle of refraction .theta..sub.2 and the thickness d of the substrate known, the physical path travelled by the transmitted beam in the substrate l.sub.s can be calculated from Equation 6.
The other distance of interest is the distance replaced by traversal of the substrate l.sub.0 which will be required to compute the increase in distance caused by insertion of the substrate in the interferometer. This distance l.sub.0 is calculated using the geometry details shown in FIG. 4 using the following equation. EQU l.sub.0 =dcos(.theta..sub.1 -.theta..sub.2)cos(.theta..sub.2) (7)
The second effect on the change in distance traversed by the transmitted beam E.sub.tr arises from the physics of propagation of an electromagnetic wave through a medium. The medium has the effect of slowing the speed of propagation of the wave to a value less than that of the speed of light in a vacuum. The index of refraction n is the key variable describing the effect. The index of refraction n is defined as the ratio of the speed v of propagation in a medium relative to the speed of light c in a vacuum. Continuity of field values at the interface of the two media ensures that the frequency of the waves is the same in both media. The net increase of optical effective distance propagated by an electromagnetic wave in a medium is given by EQU .sigma..sub.2 =n.sub.2 .sigma..sub.1 ( 8)
Thus, the distance traversed along an optical path is increased in proportion to the index of refraction of the medium through which the electromagnetic wave travels.
The optical distance z.sub.so added by the incorporation of substrate 28 in interferometer 20 is given in terms of wavelengths in the medium by the following ##EQU5##
The increased optical distance may be expressed in terms of the index of refraction n.sub.2 of the medium, the thickness d of the medium, and the angle of refraction .theta..sub.2, taking into consideration the two traversals in the model interferometer, as follows EQU z.sub.s =2n.sub.2 dcos(.theta..sub.2)-2dcos(-.theta..sub.2 +.theta..sub.1)cos(.theta..sub.2) (10)
This increase in optical distance adds one term to the primary interferometer equation derived for a simple interferometer model as expressed in Eq. 4. It can be shown that the substrate introduces an effective distance term to the angle of the cosine that describes the interferogram function that is measured by the square law detector. This added term is a function of the index of refraction of the substrate and is given by the following EQU P.sub.DS =P.sub.DS cos(2.pi..sigma.[x-z.sub.5 ]) (11)
FIG. 5 shows an interferogram for an interferometer with a substrate. A comparison of FIG. 5 with FIG. 2 shows that the effect of the substrate is to shift the phase of the cosine waveform by the distance added by the substrate (0.00010154 cm for the illustrated example). It should be noted that the amplitude and frequency of the electromagnetic wave have not been affected. In reality, the amplitude is affected by reflection and refraction losses, however, these phenomena are not crucial to the present considerations. The actual phase shift from a typical substrate having a thickness on the order of 0.5 cm is much larger (5,000 times larger) over many full cycles of the waveform (approximately 507 cycles in the illustrated example).
Thus far only monochromatic electromagnetic radiation has been considered. Practical use of interferometers employs broadband radiation having wavelengths extending over a finite range of wavenumber values. The superposition principle of electromagnetic waves provides that each wavelength may be analyzed individually and the results summed to predict the resulting waveform present at the detector during a scan (interferogram). This means that the results for the models thus far analyzed are valid and a direct integration over the bandwidth of interest may be performed by replacing the power terms with appropriate power density functions of wavenumber.
Summing (or integrating) over the bandwidth of interest is given by the following equation ##EQU6##
The power at the detector is replaced by the interferogram function I(x) which is the detector response as a function of retardation generated by scanning the moving retroreflector. The incident wave's power at a specific wavenumber has been replaced by a spectrum function S (wavenumber) which is the power per wavenumber over an infinitesimally small wavenumber interval.
The object of interferometric spectroscopy is to determine the spectrum by measuring the interferogram which is defined by Eq. 12. Well known methods are available for calculating the spectrum from the measured interferogram for the case of broadband radiation with no substrate present. These techniques make use of the relationships between Fourier-transform pairs to determine the spectrum, with Eq. 12 expressing the interferogram function I(x) in the form of a cosine Fourier-transform pair. If an ideal interferometer (without substrate) is used to measure the interferogram, a cosine Fourier-transform is all that is required to calculate the desired spectrum as set forth in Eq. 13. ##EQU7##
When the effects of the substrate are taken into account, the simple cosine integral has an additional phase term Z.sub.s .sigma. as set forth in the following EQU Z.sub.s =n.sub.s dcos(.theta..sub.2)-dcos(.theta..sub.1 -.theta..sub.2)cos(.theta..sub.2) (14)
The phase shift is a function of wavenumber .theta. and thus depends upon the index of refraction of the substrate material. However, the index of refraction is not constant over the spectral range of interest as shown in FIG. 6 which is a graphic representation of the variation of index of refraction of potassium bromide (KBr) over the spectral range of interest (400 to 4,000 cm.sup.-1).
If the index of refraction were constant over the spectral range of interest, the linear term in the phase shift could be easily corrected by a mere shift of the X-axis in FIG. 5. Eq. 14 indicates that z.sub.s would be constant (n being constant), hence the term "x-z.sub.s " could be simplified by defining a new independent variable as y=(x-z.sub.s). The simple cosine Fourier analysis would again be sufficient to determine the desired spectrum. The physical interpretation of this axes shift is simply that the point of zero path difference (ZPD) of the interferogram scan is simply shifted by the length of the optical path added to the substrate (z.sub.s).
The additional phase term that depends on wavenumber complicates the Fourier analysis. However, with proper care the spectrum may be reliably determined. The solution uses the identity that allows the cosine of the sum of two angles to be expanded as shown in Eq. 15. The expanded form of the cosine term can be used to express the interferogram as two separate sine and cosine integrals as set forth in Eq. 16. The sine and cosine forms indicate that a complex Fourier analysis will be needed to obtain the desired spectrum from the interferogram. ##EQU8##
To solve the problem, the interferogram must be divided into two components, a symmetric component I.sub.s and an asymmetric component I.sub.A. From the form of the integrals and their dependence on x, we can see that the first term in Eq. 16 is symmetric with respect to x and that the second integral is asymmetric with respect to x. This allows Eq. 16 to be rewritten as two separate equations, where the Fourier-transform pair can be identified by treating the spectrum multiplied by the phase component as a single spectral function to be Fourier analyzed. In this manner, the cosine Fourier-transform of the symmetric portion of the interferogram and the sine Fourier-transform of the asymmetric portion of the interferogram can be calculated respectively by the following two equations ##EQU9##
For simplicity, the sign and 1/2 .pi. factors that enter into the inverse Fourier-transform expressions have been omitted.
Eqs. 17 and 18 are nearly in a form to provide the desired spectrum as a well defined calculation based upon the measured interferogram. However, there are significant practical problems associated with the desired solution. One is associated with the matter of determining the symmetric and asymmetric portions of the interferogram related to the problem of precisely determining the point where the retardation is zero. The second problem is associated with determining the phase term to sufficient accuracy. These two problems are related in that if there is no phase term, the zero path difference (ZPD) could easily be determined by finding a simple maxima of the interferogram function. However, the phase term is a very large number, typically several hundreds or even thousands of cycles (6.28 radians per cycle), which becomes significant when considering that even a small fraction of a cycle (0.1 radian) phase shift can introduce spectral errors as large as 100%.
The necessity of the substrate and the interference of the resulting phase shift has plagued interferometers from their initial designs. A solution to this problem taken by Michelson involved the introduction of an optical compensating element in the opposing arm of the interferometer, or in the path that does not go through the substrate. A conventional interferometer with a compensator 30 is shown schematically in FIG. 7. Compensator 30 is an optical element identical to the substrate 28, comprised of the same material and having the same thickness. The positioning of the compensator 30 requires the reflected beam E.sub.rf to undergo the same refraction and dispersion present in the transmitted beam E.sub.tr. This optical solution was used for interferometric applications from the time of the invention of the Michelson interferometer until 1969. The limitation of this approach involving the impossibility of matching the thickness and optical properties of the substrate and compensator to completely correct the dispersion of the substrate limited the accuracy of the measured results. This approach also suffered from other small phase error terms that arise from the electronics used to process the interferogram waveform, the detector's frequency response characteristics, the discrete sampling techniques (analogue-to-digital conversion), as well as the residual phase imbalance between the compensator and substrate.
The present invention addresses the aforementioned limitations of the prior art by eliminating the compensator in a Michelson interferometer and avoiding the considerations and complexity associated therewith. By eliminating the interferometer compensator, reflection and beamsplitter edge losses are reduced, interferometer signal dynamic range is limited for reduced signal processing requirements, precise interferometer alignment and dimensional criteria may be relaxed, and interferometer cost is reduced by the reduced number of components and complexity.